2. Linear Equations and Inequalities

Absolute Value Inequalities

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Multiple Choice

Solve and graph the following absolute value inequalities.
(A) $∣ 8 x + 6 ∣ + 5 < 4$

All real numbers

Number line from -8 to 8 with an arrow indicating the positive x-direction.

$\left\lbrack-1\frac78,-\frac38\right\rbrack$

Number line from -8 to 8 with red brackets highlighting the interval between -2 and -1.

$\left(-1\frac78,-\frac38\right)$

Number line from -8 to 8 with a red open interval between -1 and 0 highlighted.

No solution

Verified step by step guidance

Start with the given inequality: $\left|8x + 6\right| + 5 < 4$.

Isolate the absolute value expression by subtracting 5 from both sides: $\left|8x + 6\right| < 4 - 5$.

Simplify the right side: $\left|8x + 6\right| < -1$.

Recall that the absolute value of any expression is always greater than or equal to zero, so it can never be less than a negative number.

Conclude that there is no solution to this inequality because $\left|8x + 6\right|$ cannot be less than $-1$.

Master Solve Absolute Value Inequalities of the Form |X|<a with a bite sized video explanation from Patrick Ford

Solve and graph the following absolute value inequalities.(A) ∣8x+6∣+5<4\left|8x+6\right|+5<4